# Project

Introduction, problems, and more.

## Introduction

Symmetry is one of the most important notions in mathematics. In many situations, typically when studying Physics problems, the objects under consideration have many symmetries. An extremely important case, with far reaching applications, is when the symmetries can be described as a family parametrized by a smooth manifold, called Lie group. A common example is the motions in the Euclidean plane/space. Among the nice features of Lie groups is the fact that they can be studied, through their corresponding Lie algebras, by using linear algebra methods. Lie groups and Lie algebras are fundamental notions, and belong to the toolkit of any research mathematician and theoretical physicist.

The construction of these Lie algebras is elementary, using only the definition of graphs and wedge products of elements of a vector space. Due to fact that they provide a link between simple graphs on one hand and geometry of Lie groups on the other, these Lie algebras have recently enjoyed a strong interest among the mathematicians. The constructed Lie algebras are 2-step nilpotent, and were first introduced in S. Dani, M. Mainkar, “Anosov aoutomorphisms on compact nil-manifolds associated with graphs”, Trans. Amer. Math. Soc. 357(2004) 2235 -2251. Recently, in M. Mainkar, “A short note on a 2-step nilpotent Lie algebras associated with graphs”, arxiv:1010.40055, it was proved that the construction has a nice property: two simple graphs are isomorphic if, and only if, the corresponding Lie algebras are isomorphic. In addition to that, this construction leads to nil-manifolds with Anosov diffeomorphisms (see the former paper above), and a characterization of graphs for which the corresponding Lie algebra admits a special geometric structure, a symplectic form, is known (H. Pouseele, P. Tirao, “Compact nil-manifolds associated with graphs”, J. Pure Appl. Alg. 213 (2009) 1788 -1794). In the last summer’sresearch project, a natural category of simple graphs and morphisms between them was introduced. It was shown that the construction of Lie algebras above determines a natural functor between this category and the category of two-step nilpotent Lie algebras. A study of the image of that functor was undertaken. The main goal was to establish a dictionary between the properties of simple graphs and (linear-) algebraic properties of the corresponding Lie algebras. A proposed interpretation of paths and cycles in terms of linear algebra terms was shown to work effectively in the case of star-type trees, and a characterization of Lie algebras coming from such graphs was proved.

## Warm-up

Find further interpretations of properties of graphs in algebraic terms, via their corresponding Lie algebras. Find an intrinsic characterization of the Lie algebras coming from trees. Besides the trees, graphs of special interest here are the bipartite, Eulerian, Hamiltonian, and k-connected graphs.

## The Setup

The second problem is related to geometric structures which arise from graphs (in the sense mentioned above). The nilpotent Lie groups and algebras provide an important source of examples (and counterexamples) for interesting geometric structures. These include complex, hyper-complex, para-hyper-complex, and hyper-symplectic structures, and are easily described (locally on a manifold) in terms of linear algebra. The interest in these structures within the mathematics community is strong because of their physics applications. The PD is involved in an ongoing research studying para-hyper-complex structures (J.Davidov, G.Grantcharov, O.Muskarov, M.Yotov, “Para-hyper-hermitian surfaces”, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 52(100) (2009), no. 3, 281–289. , and J.Davidov, G.Grantcharov, O.Muskarov, M.Yotov, “Generalized Pseudo Kahler structures”, Comm. Math. Phys.(to appear)). For all types of structures described above some of the first examples (or counterexamples) are based on nilpotent Lie groups. For instance, one of the first complex non-Kahler manifolds was provided by a 4-dimensional nilpotent Lie group. In fact, the groups and algebras in question are especially appropriate to test the properties of the corresponding geometric structures, and are more effective compared to the “standard” methods used in the area. In the last summer’s project,our students found two infinite series of graphs whose Lie algebras admit integrable complex structures. Based on their findings, we formulate the following

## Main Problem

Find the graphs for which the corresponding Lie algebra carries one of the following structures: complex, hyper-complex, para-hyper-complex, or hyper-symplectic. Look for examples of Lie algebras, coming from graphs, with (almost)complex structures with few holomorphic functions.